Schreier refinement theorem

In mathematics, the Schreier refinement theorem of group theory states that any two subnormal series of subgroups of a given group have equivalent refinements, where two series are equivalent if there is a bijection between their factor groups that sends each factor group to an isomorphic one.

The theorem is named after the Austrian mathematician Otto Schreier who proved it in 1928. It provides an elegant proof of the Jordan–Hölder theorem. It is often proved using the Zassenhaus lemma. Baumslag (2006) gives a short proof by intersecting the terms in one subnormal series with those in the other series.

ExampleEdit

Consider  , where   is the symmetric group of degree 3. There are subnormal series

 
 

  contains the normal subgroup  . Hence these have refinements

 

with factor groups isomorphic to   and

 

with factor groups isomorphic to  .

ReferencesEdit

  • Baumslag, Benjamin (2006), "A simple way of proving the Jordan-Hölder-Schreier theorem", American Mathematical Monthly, 113 (10): 933–935, doi:10.2307/27642092